Properties

Label 1077.41
Modulus $1077$
Conductor $1077$
Order $358$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1077, base_ring=CyclotomicField(358))
 
M = H._module
 
chi = DirichletCharacter(H, M([179,60]))
 
pari: [g,chi] = znchar(Mod(41,1077))
 

Basic properties

Modulus: \(1077\)
Conductor: \(1077\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(358\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1077.g

\(\chi_{1077}(2,\cdot)\) \(\chi_{1077}(5,\cdot)\) \(\chi_{1077}(8,\cdot)\) \(\chi_{1077}(11,\cdot)\) \(\chi_{1077}(17,\cdot)\) \(\chi_{1077}(20,\cdot)\) \(\chi_{1077}(23,\cdot)\) \(\chi_{1077}(32,\cdot)\) \(\chi_{1077}(41,\cdot)\) \(\chi_{1077}(44,\cdot)\) \(\chi_{1077}(47,\cdot)\) \(\chi_{1077}(50,\cdot)\) \(\chi_{1077}(68,\cdot)\) \(\chi_{1077}(74,\cdot)\) \(\chi_{1077}(80,\cdot)\) \(\chi_{1077}(92,\cdot)\) \(\chi_{1077}(98,\cdot)\) \(\chi_{1077}(101,\cdot)\) \(\chi_{1077}(107,\cdot)\) \(\chi_{1077}(110,\cdot)\) \(\chi_{1077}(125,\cdot)\) \(\chi_{1077}(128,\cdot)\) \(\chi_{1077}(131,\cdot)\) \(\chi_{1077}(146,\cdot)\) \(\chi_{1077}(149,\cdot)\) \(\chi_{1077}(158,\cdot)\) \(\chi_{1077}(164,\cdot)\) \(\chi_{1077}(170,\cdot)\) \(\chi_{1077}(173,\cdot)\) \(\chi_{1077}(176,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{179})$
Fixed field: Number field defined by a degree 358 polynomial (not computed)

Values on generators

\((719,7)\) → \((-1,e\left(\frac{30}{179}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1077 }(41, a) \) \(-1\)\(1\)\(e\left(\frac{45}{358}\right)\)\(e\left(\frac{45}{179}\right)\)\(e\left(\frac{99}{358}\right)\)\(e\left(\frac{30}{179}\right)\)\(e\left(\frac{135}{358}\right)\)\(e\left(\frac{72}{179}\right)\)\(e\left(\frac{193}{358}\right)\)\(e\left(\frac{64}{179}\right)\)\(e\left(\frac{105}{358}\right)\)\(e\left(\frac{90}{179}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1077 }(41,a) \;\) at \(\;a = \) e.g. 2